I tested Oracle Java 1.8 u131 as well as 1.6 u64 in Firefox 51.0.1 as well as Iceweasel / Firefox/3.5.16 in a chroot.

Using linux-image-4.4.0-81-generic it crashes in all combinations while with both other kernels it works.

I was not able to obtain any detailed crash information from Firefox 51.0.1, but Iceweasel 3.5.16 crashed completely, allowing me to obtain a stack trace which shows the relation to stack operations performed by the plugin, even without proper debug symbols:

I'm running the Ubuntu 16.04 based KDE Neon distribution which somehow apparently does not allow me to use apport to report this bug:

> $ LANG= apport-cli linux-image-4.4.0-81-generic
>
> *** Collecting problem information
>
> The collected information can be sent to the developers to improve the
> application. This might take a few minutes.
> .........
>
> *** Problem in linux-image-4.4.0-81-generic
>
> The problem cannot be reported:
>
> This is not an official KDE package. Please remove any third party package and try again.

If someone can tell me how to get apport working for this package, I can use it to collect additional information, but (unfortunately?) the problem should be fairly easy to reproduce...

The same bug appears on 17.04. 4.10.0-24-generic, which contains CVE-2017-1000364 fix, causes Oracle java plugin to crash. 4.10.0-22-generic, in exactly the same setup, works fine. The console error from Firefox (what most users will see) is

This is a ***MASSIVE REGRESSION*** affecting many or even all native applications that use the Java Invocation API, including at least Eclipse (crashes a few seconds after startup), and LibreOffice Base with any JDBC database connector (instant crash as soon as it tries to load the JVM).

Scilab is affected too. It uses openjdk-8.
See bug 1699892 for details.
Scilab is crashing with new kernel (linux-image-3.13.0-121-generic in Trusty / linux-image-4.4.0-81-generic in Xenial), but works with previous one (linux-image-3.13.0-119-generic in Trusty / linux-image-4.4.0-78-generic in Xenial).

John Johansen:
"The kernel team is aware of the issue, and will be releasing updated
kernels when they are available.

There are currently no plans to revert the kernel patch until the
replacement patches are ready due to the nature of the security
vulnerability. If the regression is preventing you from using the
applications you require then we currently recommend you reboot into
the previous kernel."

I can reliably reproduce the crash in an Ubuntu 17.04 x86 virtual machine by ensuring that java is enabled in libreoffice's advanced options (using the openjdk-8 package), launching base and creating a new database.